Maximal torus, character lattice, and weights

A torus over a field $k$ is an algebraic $k$ -group $T$ such that $T_{\overline k}\simeq\mathbb G_m^r$ over an algebraic closure. It is split if this isomorphism already exists over $k$ . A maximal torus $T\subset G$ is a torus not properly contained in another torus of $G$ .

The geometric character and cocharacter lattices are

X^*(T)= \operatorname{Hom}_{\overline k\text{-grp}} (T_{\overline k},\mathbb G_m), \qquad X_*(T)= \operatorname{Hom}_{\overline k\text{-grp}} (\mathbb G_m,T_{\overline k}).

They are dual finite free abelian groups with a natural action of the absolute Galois group $\Gamma_k$ . The characters defined over $k$ are $X^*(T)^{\Gamma_k}$ , not generally all of $X^*(T)$ .

Weights

For a representation of $G_{\overline k}$ , its weights are characters in $X^*(T)$ . The abstract weight lattice of the root system can be larger than the actual character lattice of $T$ ; the intermediate lattice records the isogeny form of a semisimple group.

Example

For the diagonal torus in $\operatorname{GL}_n$ ,

X^*(T)\simeq\mathbb Z^n, \qquad (m_i)\longmapsto \left(\operatorname{diag}(t_i)\mapsto\prod_i t_i^{m_i}\right).

Langlands role

The full based root datum uses both $X^*(T)$ and $X_*(T)$ . Exchanging them, and roots with coroots, defines the Langlands dual group.

References

A. Borel, Linear Algebraic Groups, second edition, Springer, 1991.